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Using equation 10.6, write the differential equation for V.

$饾啅{鈭�}^{2}V+\frac{鈭�}{鈭�t}=-\frac{1}{{蔚}_{鈭�}}p$ 鈥︹�� (1)

Here.$饾啅$ is d鈥� Alembertian.

Similarly, write the differential equation for A.

${饾啅}^2\mathit{A}\mathbf{-}鈭�L=-{渭}_{鈭�}\mathit{J}$

Substitute ${饾啅}^2={鈭�}^2-{渭}_{鈭�}{蔚}_{鈭�}\frac{{鈭�}^2}{鈭�t^2}\mathbf{and}\mathbf{}\mathrm{L}=鈭�.\mathbf{A}+{\mathrm{渭}}_{鈭�}{\mathrm{蔚}}_{鈭�}\frac{鈭�\mathrm{V}}{鈭�\mathrm{t}}$and in equation (1).

$$\begin{gathered} \left({饾啅}^2-{渭}_{鈭�}{蔚}_{鈭�}\frac{{鈭�}^2}{鈭�t^2}\right)V+\frac{鈭�}{鈭�t}\left(\mathbf{鈭�}\mathbf{.}\mathbf{A}+{\mathrm{渭}}_{鈭�}{\mathrm{蔚}}_{鈭�}\frac{鈭�\mathrm{V}}{鈭�\mathrm{t}}\right)=-\frac{1}{{\mathrm{蔚}}_{鈭�}}p \\ {鈭�}^{2}V-{渭}_{鈭�}{蔚}_{鈭�}\frac{{鈭�}^{2}V}{鈭�t^2}+\frac{鈭�}{鈭�t}\left(鈭�.\mathbf{A}\right)+{\mathrm{渭}}_{鈭�}{\mathrm{蔚}}_{鈭�}\frac{{鈭�}^2\mathrm{V}}{鈭�{\mathrm{t}}^2}=-\frac{1}{{\mathrm{蔚}}_{鈭�}}\mathrm{p} \\ {鈭�}^2\mathrm{V}+\frac{鈭�}{鈭�\mathrm{t}}\left(鈭�.\mathbf{A}\right)=-\frac{1}{{\mathrm{蔚}}_{鈭�}}\mathrm{p} \end{gathered}$$

Which is equal to the equation 10.4 as${鈭�}^2\mathrm{V}+\frac{鈭�}{鈭�\mathrm{t}}\left(鈭�.\mathbf{A}\right)=-\frac{1}{{\mathrm{蔚}}_{鈭�}}\mathrm{p}$.

Substitute${饾啅}^2={鈭�}^2-{渭}_{鈭�}{蔚}_{鈭�}\frac{{鈭�}^2}{鈭�t^2}\mathbf{and}\mathbf{}\mathrm{L}=鈭�.\mathbf{A}+{\mathrm{渭}}_{鈭�}{\mathrm{蔚}}_{鈭�}\frac{鈭�\mathrm{V}}{鈭�\mathrm{t}}$in equation (2).

$$\begin{gathered} \left({鈭�}^2-{渭}_{鈭�}{蔚}_{鈭�}\frac{{鈭�}^2}{鈭�t^2}\right)\mathit{A}-鈭�\left(\mathbf{鈭�}\mathbf{.}\mathbf{A}+{\mathrm{渭}}_{鈭�}{\mathrm{蔚}}_{鈭�}\frac{鈭�\mathrm{V}}{鈭�\mathrm{t}}\right)=-{渭}_{鈭�}\mathit{J} \\ \left({鈭�}^2\mathit{A}-{渭}_{鈭�}{蔚}_{鈭�}\frac{{鈭�}^2\mathit{A}}{鈭�t^2}\right)-鈭�\left(鈭�.\mathit{A}+{\mathrm{渭}}_{鈭�}{\mathrm{蔚}}_{鈭�}\frac{鈭�\mathrm{V}}{鈭�\mathrm{t}}\right)=-{渭}_{鈭�}\mathit{J} \end{gathered}$$

Which is equal to the equation 10.5 as .

$\left({鈭�}^2\mathit{A}-{渭}_{鈭�}{蔚}_{鈭�}\frac{{鈭�}^2\mathit{A}}{鈭�t^2}\right)-鈭�\left(鈭�.\mathit{A}+{\mathrm{渭}}_{鈭�}{\mathrm{蔚}}_{鈭�}\frac{鈭�\mathrm{V}}{鈭�\mathrm{t}}\right)=-{渭}_{鈭�}\mathit{J}$

Therefore, the differential equations for V and Ain the symmetrical form are derived as ${鈭�}^{2}V+\frac{鈭�}{鈭�t}\left(鈭�.\mathit{A}\right)=-\frac{1}{{蔚}_{鈭�}}p$and .

$\left({鈭�}^2\mathit{A}-{渭}_{鈭�}{蔚}_{鈭�}\frac{{鈭�}^2\mathit{A}}{鈭�t^2}\right)-鈭�\left(鈭�.\mathit{A}+{\mathrm{渭}}_{鈭�}{\mathrm{蔚}}_{鈭�}\frac{鈭�\mathrm{V}}{鈭�\mathrm{t}}\right)=-{渭}_{鈭�}\mathit{J}$